How to Round Numbers; Preliminaries: On Algorithms Involving Numbers; Diophantine Approximation, Problems; Lattices, Bases, and the Reduction Problem; Diophantine Approximation and Rounding; What is a Real Number How to Round a Convex Body; Preliminaries: Inputting a Set; Algorithmic Problems on Convex Sets; The Ellipsoid Method; Rational Polyhedra; Some Other Algorithmic Problems on Convex Sets; Integer Programming in Fixed Dimension; Some Applications in Combinatorics; Cuts and Joins; Chromatic Number, Cliques and Perfect Graphs; Minimizing a Submodular Function.